# Equation Of A Line Concise Solutions Chapter-14 Class 10

Equation Of A Line Concise Solutions Chapter-14 Class 10. Solutions of Exercise – 14 (A), Exercise – 14 (B),Exercise – 14 (C), Exercise – 14 (D),Exercise – 14 (E) of Concise.  Concise Selina Maths of ICSE Board Class 10th Solutions for Equation Of A Line Concise Solutions Chapter-14. Step by Step Solutions of Concise Equation Of A LineChapter-14 for ICSE Maths Class 10 is available here. All Solutions of Concise Selina of Chapter-14 Equation Of A Line has been solved according instruction given by council. This is the  Solutions of Chapter-14 Equation Of A Line for ICSE Class 10th. ICSE Maths text book of Concise is In series of famous ICSE writer in maths publications. Concise is most famous among students.

## Equation Of A Line Concise Solutions Chapter-14 Class 10

The Solutions of Concise Mathematics Equation Of A Line Concise Solutions Chapter-14 for ICSE Class 10 have been solved.  Experience teachers Solved Equation Of A Line Concise Solutions Chapter-14 to help students of class 10th ICSE board. Therefore the ICSE Class 10th Maths Solutions of Concise Selina Publishers helpful on  various topics which are prescribed in most ICSE Maths textbook

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Exe-14(A),    Exe-14(B),    Exe-14(C)   Exe-14(D),    Exe-14(E)

### How to Solve Concise Maths Selina Publications Chapter-14 Equation Of A Line

Note:- Before viewing Solutions of Chapter -14 Equation Of A Line of Concise Selina Maths. Read the Chapter Carefully then solve all example of your text bookThe Chapter- 14 Equation Of A Line is main Chapter in ICSE board.

### Concise Solutions of Exercise -14(A),Equation of a line for ICSE Class 10th Maths

#### Question 1.

Find, which of the following points lie on the line x – 2y + 5 = 0 :
(i) (1, 3)
(ii) (0, 5)
(iii) (-5, 0)
(iv) (5, 5)
(v) (2, -1.5)
(vi) (-2, -1.5)

##### (i)

Substituting x = 1, y = 3, in the given equation.
1 – 2 x 3 + 5 = 0 ⇒ 1 – 6 + 5 = 0 ⇒ 0 = 0, which is true.
(1, 3) satisfies the equation.
(ii)

Substituting x = 0 , y = 5 in the given equation
0 – 2 x 5 + 5 = 0 ⇒ 0 – 10 + 5 = 0 ⇒ -5 = 0, which is not true.
( 0, 5) does not satisfy the equation.

##### (iii)

Substituting x = – 5, y = 0 in the given equation
-5 – 2 x 0 + 5 = 0 ⇒ -5 – 0 + 5 = 0 ⇒ 0 = 0 which is true.
(-5, 0) satisfies the equation.

##### (iv)

Substituting x = 5, y = 5 in the given equation.
– 5 – 2 x 5 + 5 = 0 ⇒ -5 – 10 + 5 = 0 ⇒ 0 = 0 which is true.
(5, 5) satisfies the equation.

##### (v)

Substituting x = 2, y = -1.5 in the given equation.
2 – 2 x (- 1.5) + 5 = 0 ⇒ 2 + 3 + 5 = 0 ⇒ 10 = 0. which is not true.
(2, -1.5) does not satisfy the equation.

##### (vi)

Substituting x = -2, y = -1.5 in the given equation
– 2 – 2 x (-1.5) + 5 = 0 ⇒ – 2 + 3 + 5 = 0 ⇒ 6 = 0, which is not true.
(-2, -1.5) does not satisfies the equation.

#### Question 2

State, true or false :
(i) the line $\frac { x }{ 2 }$ + $\frac { y }{ 3 }$ = 0 passes through the point (2, 3).
(ii) the line $\frac { x }{ 2 }$ + $\frac { y }{ 3 }$ = 0 passes through the point (4, -6).
(iii) the point (8, 7) lies on the line y – 7 = 0
(iv) the point (-3, 0) lies on the line x + 3 = 0
(v) if the point (2, a) lies on the line 2x – y = 3, then a = 5.
Solution:

##### (i)

Equation of the line is $\frac { x }{ 2 }$ + $\frac { y }{ 3 }$ = 0
and co-ordinates of point are (2, 3)
If the point is on the line, then it will satisfy the equation.

(2, 3) is not on the line

##### (ii)

Equation of the line is $\frac { x }{ 2 }$ + $\frac { y }{ 3 }$ = 0
and co-ordinates of point are (4, -6)
If the point is on the line, then it will satisfy the equation

Hence, point (4, -6) is on the line.

##### (iii)

Equation of line is y – 7 = 0 and the co-ordinates of point are (8, 7)
If the point is on the line, then it will satisfy the equation
L.H.S. = y – 7 = 7 – 7 = 0 = R.H.S.
Hence, point (8, 7) is on the line.

##### (iv)

Equation of the line is x + 3 = 0 and co-ordinates of point are (-3, 0)
If the point is on the line, then it will satisfy the equation.
L.H.S. = x + 3 = -3 + 3 = 0 = R.H.S.
Hence, the point (-3, 0) is on the line.

##### (v)

Equation of the line is 2x – y = 3
and co-ordinates of the point are (2, a)
If the point is on the line, then it will satisfy the equation.
L.H.S. = 2x – y = 2 x 2 – a = 4 – a
R.H.S. = 3
4 – a = 3 ⇒ 4 + 3 = a ⇒ a = 7
But a = 5 given, therefore it is not on the line.
(i) False

(ii) True

(iii) True

(iv) True

(v) False.

#### Question 3

The line given by the equation 2x – $\frac { y }{ 3 }$ = 7 passes through the point (k, 6); calculate the value of k.

Question 4

For what value of k will the point (3, -k) lie on the line 9x + 4y = 3 ?

Point (3, -k) satisfies the equation 9x + 4y = 3
Substituting x = 3 , y = -k, we get :
9 x 3 + 4 (- k), = 3
⇒ 27 – 4k = 3
and ⇒ – 4k = 3 – 27
so ⇒ – 4k = – 24
hence ⇒ k = 6

#### Question 5

The line $\frac { 3x }{ 5 }$ – $\frac { 2y }{ 3 }$ + 1 = 0, contains the point (m, 2m – 1); calculate the value of m.

Point (m, 2m -1) satisfies the equation

#### Does the line 3x – 5y = 6 bisect the join of (5, -2) and (-1, 2) ?

Line 3x – 5y = 6 bisect the join of points (5, -2) and (-1, 2)
The mid-point of (5, -2) and (-1, 2) satisfies the equation.
Now, mid-point of (5, -2) and (-1, 2)

Now, substituting x = 2, y = 0, in the given equation

#### Question 7

(i) The line y = 3x – 2 bisects the join of (a, 3) and (2, -5), find the value of k.
(ii) The line x – 6y + 11 = 0 bisects the join of (8, -1) and ( 0, k). Find the value of k.

(i) line y = 3x – 2 bisects the join of (a, 3) and (2, -5)
Mid-point join of there points satisfies it.
Now, mid-point of (a, 3) and (2, -5) is

#### Question 8

(i) The point (-3, -2) lies on the line ax + 3y + 6 = 0, calculate the value of ‘a’
(ii) The line y = mx + 8 contains the point (- 4, 4), calculate the value of ‘m’

(i) Point (-3, 2) lies on the line ax + 3y + 6 = 0,
Then x = – 3, y = 2 satisfies it
a (-3) + 3(2) + 6 = 0
⇒ -3a + 6 + 6 = 0
so ⇒ -3a + 12 = 0
therefore ⇒ -3a = – 12
hence ⇒ a = 4
(ii) line y = mx + 8 contains the point (-4, 4)
x = – 4, y = 4 satisfies it
4 = m (-4) + 8
⇒ 4 = -4m + 8
and ⇒ 4m = 8 – 4 = 4
⇒ m = 1

#### Question 9

The point P divides the join of (2, 1) and (-3, 6) in the ratio 2 : 3. Does P lie on the line x – 5y + 15 = 0 ?

P divides the line joining of the points (2, 1) and (-3, 6) in the ratio of 2 : 3,
co-ordinates of P will be

Now, substituting x = 0, y = 3 in the equation
x – 5y + 15 = 0
⇒ 0 – 5 x 3 + 15 = 0
and ⇒ 0 – 15 + 15 = 0
⇒ 0 = 0 which is true.
Point (0, 3) lies on the line.

#### Question 10

The line segment joining the points (5, -4) and (2, 2) is divided by the point Q in the ratio of 1 : 2. Does the line x – 2y = 0 contain Q ?

Point Q, divides the line segment joining the points (5, -4) and (2, 2) in the rates of 1 : 2
co-ordinates of Q will be,

Now, substituting x = 4, y = – 2 in the equation
x – 2y = 0, we get
4 – 2 x (-2) = 0
⇒ 4 + 4 = 0
⇒ 8 = 0 which is not true.
Point Q does not lie on the line x – 2y = 0

#### Question 11

Find the point of intersection of the lines : 4x + 3y = 1 and 3x – y + 9 = 0. If this point lies on the line (2k – 1) x – 2y = 4; find the value of k.

4x + 3y = 1 …..(i)
3x – y = -9 …..(ii).
Multiplying (i) by 1 and (ii) 3
4x + 3y = 1
9x – 3y = -27
1 3x = – 26 ⇒ x = -2
from (ii),
3x – y = – 9
3(-2) – y = -9
⇒ – 6 – y = -9
and ⇒ -y = -9 + 6 = -3
then ⇒ y = 3
The point of intersection is (-2, 3)
The line (2k – 1) x – 2y = 4 passes through that point also
It is satisfy it.
(2k – 1) (-2) – 2(3) = 4
⇒ -4k + 2 – 6 = 4
and ⇒ -4k – 4 = 4
so ⇒ -4k = 4 + 4 = 8
therefore ⇒ k = -2
Hence point of intersection is (-2, 3) and value of k = -2

#### Question 12

Show that the lines 2x + 5y = 1, x – 3y = 6 and x + 5y + 2 = 0 are concurrent.

2x + 5y = 1, x – 3y = 6 and x + 5y + 2 = 0 are concurrent
They will pass through the same point
Now 2x + 5y = 1 …..(i)
x – 3y = 6 ……(ii)
Multiply (i) by 3 and (ii) by 5, we get :
-6x + 15y = 3
5x – 15y = 30
11x = 33 ⇒ x = 3
from (ii),
x – 3y = 6
⇒ 3 – 3y = 6
and ⇒ -3y = 6 – 3 = 3
hence ⇒ y = -1
Point of intersection of first two lines is (3, -1)
Substituting the values in third line x + 5y + 2 = 0
L.H.S. = x + 5y + 2 = 3 + 5(-1) + 2 = 3 – 5 + 2 = 5 – 5 = 0 = R.H.S.
Hence the given three lines are concurrent.

### Concise Solutions of EXERCISE -14 (B), Equation Of A Straight Line Selina Publishers Maths

#### Question 1

Find the slope of the line whose inclination is :
(i) 0°
(ii) 30°
(iii) 72° 30′
(iv) 46°

(i) Slope of line whose inclination is 0° = tanθ = tan 0° = 0
(ii) Slope of line whose inclination is 30° = tan 30° = $\frac { 1 }{ \surd 3 }$
(iii) Slope of line whose inclination is 72° 30′ = tan 72°30’ = 3. 1716 (Using tables)
(iv) Slope of line whose inclination is 46° = tan 46° = 1.0355 (Using tables)

#### Question 2

Find the inclination of the line whose slope is:
(i) 0
(ii) √3
(iii) 0.7646
(iv) 1.0875

Slope of a line = tanθ. Where θ is the inclination
(i) When slope is θ. then tanθ = 0 ⇒ θ = 0°
(ii) When slope is θ, then tanθ = √3 ⇒ θ = 60°
(iii) When slope is 0.7646, then tanθ = 0.7646 ⇒ θ = 37°24′ (Using tables)
(iv) When slope is 1.0875, then tanθ = 1.0875 ⇒ θ = 47°24′ (Using tables)

#### Question 3

Find the slope of the line passing through the following pairs of points :
(i) (-2, -3) and (1, 2)
(ii) (-4, 0) and origin
(iii) (a, -b ) and (b, -a)

We know that, slope of a line which passes

#### Question 4

Find the slope of the line parallel to AB if:
(i) A = (-2, 4) and B = (0, 6)
(ii) A = (0, -3) and B = (-2, 5)

#### Question 5

Find the slope of the line perpendicular to AB if:
(i) A = (0, -5) and B = (-2, 4)
(ii) A = (3, -2) and B = (-1, 2)

#### Question 6

The line passing through (0, 2) and (-3, -1) is parallel to the line passing through (-1, 5) and (4, a). Find a.

Slope of the line passing through two points (0, 2) and (-3, -1)

#### Question 7

The line passing through (-4, -2) and (2, -3) is perpendicular

Slope of the line passing through the points

#### Question 8

Without using the distance formula, show that the point A (4, -2), B (-4, 4) and C (10, 6) are the vertices of a right-angled triangle.

AB and CA are perpendicular to each other
Hence, ΔABC is a right-angled triangle.

#### Question 9 Equation Of A Line Concise Solutions

Without using the distance formula, show that the points A (4, 5), B (1, 2), C (4, 3) and D (7, 6) are the vertices of a parallelogram.

Slopes of AB and DC are equal
AB || DC Similarly slope of BC and slope of DA are equal.
BC || DA
Hence ABCD is a parallelogram.

#### Question 10

(-2, 4), (4, 8), (10, 7) and (11, -5) are the vertices of a quadrilateral. Show that the – quadrilateral, obtained on joining the mid-points of its sides, is a parallelogram.

QR || PS.
Hence PQRS is a parallelogram.

#### Question 11 Equation Of A Line Concise Solutions

Show that the points P (a, b + c), Q (b, c + a) and R (c, a + b) are co llinear.

The given points are P (a, b + c), Q (b, c + a) and R (c, a + b)
We know that, these points P, Q, R are collinear if Slope of PQ = Slope of QR

Slope of PQ = Slope of QR
P, Q and R are collinear.

#### Question 12

Find x, if the slope of the line joining (x, 2) and (8, -11) is $\frac { -3 }{ 4 }$.

Slope of line joining (x, 2) and (8, -11) is

#### Question 13

The side AB of an equilateral triangle ABC is parallel to the x-axis. Find the slopes of all its sides.

#### Answer 13 Equation Of A Line Concise Solutions

ΔABC is an equilateral

Each angle is equal to 60°
Side AB is parallel to x-axis
Slope of AB = slope of x-axis = 0.
and Slope of AC = tan A = tan 60° = √3
therefore Slope of CB = tan B = tan 120° = tan (180°- 60°) = – tan 60° = -√3
hence Slopes of AB, BC and CA are 0, -√3, √3

#### Question 14

The side AB of a square ABCD is parallel to the x-axis. Find the slopes of all its sides. Also, find :
(i) the slope of the diagonal AC
(ii) the slope of the diagonal BD.

ABCD is a square in which AB || DC || x-axis.
Slope of AB and DC = 0
and slope of AD and BC = not defined (tan90° is not defined)
AC and BD are the diagonals of square ABCD.
Now slope of AC = tan 45° = 1
and slope of BD = tan 135° = tan (180° – 45°) = – tan 45° = -1

#### Question 15

A (5, 4), B (-3, -2) and C (1, -8) are the vertices of a triangle ABC. Find :
(i) the slope of the altitude of AB
(ii)  slope of the median AD and
(iii) the slope of the line parallel to AC.

Vertices of ΔABC are A (5, 4), B (-3, -2), and C (1, -8)

#### Question 16

The slope of the side BC of a rectangle ABCD is $\frac { 2 }{ 3 }$. Find
(i) The slope of the side AB,
(ii) the slope of the side AD.

ABCD is a rectangle in which

#### Question 17 Equation Of A Line Concise Solutions

Find the slope and the inclination of the line AB if
(i) A = (-3, -2) and B = (1, 2)
(ii) A = (0, -√3) and B = (3, 0)
(iii) A = (-1, 2√3) and B = (-2, √3)

#### Question 18

The points (-3, 2), (2, -1) and (a, 4) are collinear, Find ‘a’.

Points are collinear.
Slope of (-3, 2) and (2, -1) = Slope of (2, -1) and (a, 4)
Now, Slope of (-3, 2) and (2, -1) will be

#### Question 19

The points (k, 3), (2, -4) and (-k + 1, -2) are collinear. Find k.
Solution:

Points (k, 3), (2, -4) and (-k + 1, -2) are collinear
Slope of (k, 3) and (2, -4) = slope of (2, -4) and (-k + 1, -2)
Now, slope of (k, 3) and (2, -4)
M

#### Question 20 Equation Of A Line Concise Solutions

Plot the points A (1, 1), B (4, 7) and C (4, 10) on a graph paper. Connect A and B, and also A and C.
Which segment appears to have the steeper slope, AB or AC ?
Justify your conclusion by calculating the slopes of AB and AC.

#### Find the value(s) of k so that PQ will be parallel to RS. Given :

(i) P (2, 4), Q (3, 6), R (8, 1) and S (10, k)
(ii) P (3, -1), Q (7, 11), R (-1, -1) and S (1, k)
(iii) P (5, -1), Q (6, 11), R (6, -4k) and S (7, k²)

### Maths Solutions of Concise for EXERCISE – 14 (C), Equations Of A Straight Line

#### Question 1.

Find the equation of line whose :
(i) y-intercept = 2 and slope = 3,
(ii) y-intercept = -1 and slope = $\frac { -3 }{ 4 }$

(i) The point whose y-intercept = 2 will be (0, 2) and slope (m) = 3.
Equation of line will be

#### Find the equation of a line whose :

(i) y-intercept = -1 and inclination = 45°
(ii) y-intercept = 3 and inclination = 30°

(i) The point whose y-intercept is -1, will be (0, -1) and inclination = 45°
Slope (m) = tan 45° = 1
Equation will be

#### Question 3.

Find the equation of the line whose slope is $\frac { -4 }{ 3 }$ and which passes through (-3, 4).

Slope of the line (m) = $\frac { -4 }{ 3 }$
The point from which the line passes (-3, 4)
Equation of line will be y – y1 = m (x – x1)

#### Question 4. Equation Of A Line Concise Solutions

Find the equation of a line which passes through (5, 4) and makes an angle of 60° with the positive direction of the x-axis.

The line passes through the point (5, 4) and angle of inclination = 60°
slope (m) = tan 60° = √3
Equation of line
y – y1 = m (x – x1)
and ⇒ y – 4 = √3 (x – 5)
hence ⇒ y – 4 = √3 x – 5√3
so ⇒ y = √3 x + 4 – 5√3

#### Question 5.

Find the equation of the line passing through:
(i) (0, 1) and (1, 2)
(ii) (-1, -4) and (3, 0)
(iii) (4, -2) and (5, 2)

Two given points are (0, 1) and (1, 2)
Slope of the line passing through these two

#### Question 6. Equation Of A Line Concise Solutions

The co-ordinates of two points P and Q are (2, 6) and (-3, 5) respectively. Find :

and (ii) The equation of PQ,
(iii) The co-ordinates of the point where PQ intersects the x-axis.

Two points P (2,-6) and Q (-3, 5) are given.

and ⇒ 5y – 30 = x – 2
so ⇒ 5y = x – 2 + 30
hence ⇒ 5y = x + 28 ….(i)
(iii) Co-ordinates of the point where PQ intersects x-axis will be = 0
substituting, the value of y in (i)
5 x 0 = x + 28 ⇒ x + 28 = 0 ⇒ x = -28
Co-ordinates of point are (-28, 0)

#### Question 7.

The co-ordinates of two points A and B are (-3, 4) and (2, -1). Find :
(i) the equation of AB
(ii) the co-ordinates of the point where the line AB intersects they- axis.

Slope of the line passing through two points A (-3, 4) and B (2, -1) will be :

Its abscissa = 0
substituting, the value of x = 0 in (i)
0 + y = 1
y = 1
Co-ordinates of point = (0, 1)

#### Question 8.

The figure given below shows two straight lines AB and CD intersecting each other at point P (3, 4). Find the equations of AB and CD.

Two lines AB and CD intersect each other at P (3, 4)
AB inclined at angle of 45° and CD at angle of 60° with the x-axis.
Slope of AB = tan 45° = 1
and slope of CD = tan 60° = √3
Now, equation of line AB will be
y – y1 = m (x – x1)
and ⇒ y – 4 = 1 (x – 3)
⇒ y – 4 = x – 3
so ⇒ y = x – 3 + 4
therefore ⇒ y = x + 1
(ii) Equation of CD will be :
y – y1 = m (x – x1)
and ⇒ y – 4 = √3 (x – 3)
so ⇒ y – 4 = √3 x – 3√3
therefore ⇒ y = √3 x – 3√3 + 4
⇒ y = √3 x + 4 – 3√3

#### Question 9.Equation Of A Line Concise Solutions

In ΔABC, A (3, 5), B (7, 8) and C (1, -10). Find the equation of the median through A.

D is mid point of BC

y – y1 = m (x – x1)
and ⇒ y + 1 = -6 (x – 4)
so ⇒ y + 1 = -6x + 24
hence ⇒ y + 6x = -1 + 24
⇒ 6x + y = 23

#### The following figure shows a parallelogram ABCD whose side AB is parallel to the x-axis, ∠A = 60° and vertex, C = (7, 5). Find the equations of BC and CD.

ABCD is a ||gm in which AB = CD || x-axis
∠A = 60° and C (7, 5)
(i) CD || AB || x-axis ,
Equation of CD will be
y – y1 = m (x – x1)
and ⇒ y – 5 = 0 (x – 7)
so ⇒ y – 5 = 0
hence ⇒ y = 5
Slope of BC = tan 60° = √3
Equation of BC will be
y – y1 = m (x – x1)
and ⇒ y – 6 = √3 (x – 7)
so ⇒ y – 6 = √3 x – 7√3
therefore ⇒ y = √3 x + 6 – 7√3

#### Question 11.

Find the equation of the straight line passing through origin and the point of intersection of the lines x + 2y = 7 and x – y = 4.

Point of intersection of two lines
x + 2y = 7 ….(i)
x – y = 4 ….(ii)
Subtracting, we get
3y = 3
y =1
Substituting, the value of y in (ii)
x – 1 = 4
⇒ x = 4 + 1 = 5
Point of intersection is (5, 1)
Slope of the line passing through origin (0, 0) and (5, 1)

Equation of line will be
y – y1 = m (x – x1)
⇒ y – 5 = $\frac { 1 }{ 2 }$ (x – 1)
and ⇒ 5y – 25 = x – 1
so ⇒ 5y = x – 1 + 25 = x + 24
therefore ⇒ 5y = x + 24

#### Question 12. Equation Of A Line Concise Solutions

In triangle ABC, the co-ordinates of vertices A, B and C are (4, 7), (-2, 3) and (0, 1) respectively. Find the equation of median through vertex A. Also, find the equation of the line through vertex B and parallel to AC.

#### Question 13.

A, B and C have co-ordinates (0, 3), (4, 4) and (8, 0) respectively. Find the equation of the line through A and perpendicular to BC.

Slope of line through A, perpendicular to BC = -(-1) = 1
Now, the equation of line through A (0, 3) is
y – y1 = m (x – x1)
y – 3 = 1 (x – 0)
and ⇒ y – 3 = x
⇒ y = x + 3

#### Question 14.

Find the equation of the perpendicular dropped from the point (-1, 2) onto the line joining the points (1, 4) and (2, 3).

Slope of the line joining the points (1, 4) and (2, 3)

Slope of line perpendicular to the above line = 1
Equation of line passing through (-1, 2)
y – y1 = m (x – x1)
⇒ y – 2 = 1 [x -(-1)]
and ⇒ y – 2 = x + 1
so  y = x + 1 + 2
therefore ⇒ y = x + 3

#### Question 15. Equation Of A Line Concise Solutions

Find the equation of the line, whose :
(i) x-intercept = 5 and y-intercept = 3
(ii) x-intercept = -4 and y-intercept = 6
(iii) x-intercept = -8 and y-intercept = -4
(iv) x-intercept = 3 and y-intercept = -6

(i) When x-intercept = 5, then point will be (5, 0)
and when y-intercept = 3, then point will be (0, 3)
Slope of the line passing through these points

#### Question 16.

Find the equation of the line whose slope is $\frac { -5 }{ 6 }$ and x-intercept is 6.

#### Question 17. Equation Of A Line Concise Solutions

Find the equation of the line with x-intercept 5 and a point on it (-3, 2).

x-intercept of the line = 5
Point = (5, 0)
Slope of the line passing through the point (-3, 2)

#### Question 18.

Find the equation of the line through (1, 3) and making an intercept of 5 on the y- axis.

The line makes y-intercept = 5
Point = (0, 5)
Slope of the line passing through the point (1, 3) and (0, 5)

Equation of the line
y – y1 = m (x – x1)
and ⇒ y – 3 = -2 (x – 1)
so ⇒ y – 3 = -2x + 2
hence ⇒ 2x + y = 2 + 3
⇒ 2x + y = 5

#### Question 19.Equation Of A Line Concise Solutions

Find the equations of the lines passing through point (-2, 0) and equally inclined to the co-ordinate axes.

(i) Slope of line AB = tan 45° = 1
Equation passing through the point (-2, 0)
y – y1 = m (x – x1)
⇒ y – 0 = 1 (x + 2)
and⇒ y = x + 2
⇒ x – y + 2 = 0
(ii) Slope of line CD = tan (-45°) = -1
Equation passing through the point (-2, 0)
y – y1 = m (x – x1)
⇒ y – 0 = -1 (x + 2)
and ⇒ y = -x – 2
so ⇒ y + x + 2 = 0
hence⇒ x + y + 2 = 0

#### Question 20. Equation Of A Line Concise Solutions

The line through P (5, 3) intersects y axis at Q.

(i) Write the slope of the line.
(ii) Write the equation of the line.
(iii) Find the co-ordinates of Q.

(i) Here θ = 45°
So, slope of the line = tanθ = tan 45° = 1
(ii) Equation of the line through P and Q is
y – 3 = 1 (x – 5)
⇒ y – x + 2 = 0
(iii) Let the coordinates of Q be (0, y)

#### Question 21. Equation Of A Line Concise Solutions

Write down the equation of the line whose gradient is $\frac { -2 }{ 5 }$ and which passes through point P, where P divides the line segment joining A (4, -8) and B (12, 0) in the ratio 3 : 1.

Slope of the line m = $\frac { -2 }{ 5 }$
P divides the line AB, whose co-ordinates are (4, -8) and (12, 0) in the ratio of 3 : 1
Co-ordinates of P be

#### Question 22.

A (1, 4), B (3, 2) and C (7, 5) are vertices of a triangle ABC. Find :
(i) the co-ordinates of the centroid of ΔABC.
(ii) the equation of a line, through the centroid and parallel to AB. [2002]

(i) Co-ordinates of vertices of ΔABC are A (1, 4), B (3, 2), C (7, 5)
and let G be the centroid of ΔABC.
Co-ordinates of G are

#### Question 23. Equation Of A Line Concise Solutions

A (7, -1), B (4, 1) and C (-3, 4) are the vertices of a triangle ABC. Find the equation of a line through the vertex B and the point F in AC; such that AP : CP = 2 : 3.

P divides AC in the ratio of 2 : 3

### Selina Concise Maths Solutions of Equations of a Straight Line ,EXERCISE – 14(D)

#### Question 1.

Find the slope and y-intercept of the line :
(i) y = 4
(ii) ax – by = 0
(iii) 3x – 4y = 5

(i) y = 4 ⇒ y = 0x + 4
Here slope = 0 and y-intercept = 4
(ii) ax – by = 0
⇒ by = ax
⇒ y = $\frac { a }{ b }$ x + 0
Here, slope = $\frac { a }{ b }$ and y-intercept = 0
(iii) 3x – 4y = 5
⇒ – 4y = 5 – 3x
and ⇒ 4y = 3x – 5
⇒ y = $\frac { 3 }{ 4 }$ x + $\frac { -5 }{ 4 }$
Here, slope = $\frac { 3 }{ 4 }$ and y- intercept = $\frac { -5 }{ 4 }$

#### Question 2.

The equation of a line is x – y = 4. Find its – slope and y-intercept. Also, find its inclination.

x – y = 4
writing the equation in form of y = mx + c
x = 4 + y
⇒ y = x – 4
Slope = 1 and y-intercept = – 4
Slope = 1
⇒ tanθ = 1
⇒ θ = 45°

#### Question 3.

(i) Is the line 3x + 4y + 7 = 0 perpendicular to the line 28x – 21y + 50 = 0 ?
(ii) Is the line x – 3y = 4 perpendicular to the line 3x – y = 7 ?

and (iii) Is the line 3x + 2y = 5 parallel to the line x + 2y = 1 ?

(iv) Determine x so that the slope of the line through (1, 4) and (x, 2) is 2.

#### Answer 3 Equation Of A Line Concise Solutions

(i) 3x + 4y + 7 = 0
Writing the equation in form of y = mx + c
4y = -3x – 7

#### Question 4.

Find the slope of the line which is parallel to:

(i) x + 2y + 3 = 0
(ii) $\frac { x }{ 2 }$ – $\frac { y }{ 3 }$ – 1 = 0

#### Question 5.

Find the slope of the line which is perpendicular to:
(i) x – $\frac { y }{ 2 }$ + 3 = 0
(ii) $\frac { x }{ 3 }$ – 2y = 4

#### Question 6.Equation Of A Line Concise Solutions

(i) Lines 2x – by + 5 = 0 and ax + 3y = 2 are parallel to each other. Find the relation connecting a and b.
(ii) Lines mx + 3y + 7 = 0 and 5x- ny – 3 = 0 are perpendicular to each other. Find the relation connecting m and n.

(i) Writing the given equations in the form of y = mx + c, we get:
-by = -2x -5
by = 2x + 5

#### Question 7.

Find the value of p if the lines, whose equations are 2x – y + 5 = 0 and px + 3y = 4 are perpendicular to each other.

Writing the given equations 2x – y + 5 = 0 and px + 3y = 4 in form of y = mx + c
2x – y + 5 = 0
-y = -2x -5
y = 2x + 5
Here, slope of the line = 2
Again, px + 3y = 4
3y = – px + 4

#### Question 8.

The equation of a line AB is 2x – 2y + 3 = 0.
(i) Find the slope of the line AB.
(ii) Calculate the angle that the line AB makes with the positive direction of the x-axis.

The line AB is given.2* – 2y + 3 = 0
Writing it in the form of y = mx + c
-2y = -2x – 3
and ⇒ 2y = 2x + 3
⇒ y = x + $\frac { 3 }{ 2 }$
Here, slope of the line = 1
Angle of inclination = tanθ
tanθ = 1
θ = 45°

#### Question 9.Equation Of A Line Concise Solutions

The lines represented by 4x + 3y = 9 and px – 6y + 3 = 0 are parallel. Find the value of p.

Writing the given lines 4x + 3y = 9 and px – 6y + 3 = 0 in the form of y = mx + c
4x + 3y = 9
⇒ 3y = – 4x + 9

#### Question 10.

If the lines y = 3x + 7 and 2y +px = 3 are perpendicular to each other, find the value of p. (2006)

#### Question 11.

The line through A (-2, 3) and B (4, 6) is perpendicular to the line 2x – 4y = 5. Find the value of b.

Gradient (mx) of the line passing through the points A (-2, 3) and B (4, b)

#### Question 12.

Find the equation of the line passing through (-5, 7) and parallel to
(i) x-axis
(ii) y-axis.

(i) Slope of the line parallel to x-axis = 0
Equation of line passing through (-5, 7) whose slope is 0.
y – 7 = 0 [x – (-5)]
and ⇒ y – 7 = 0
⇒ y = 7
(ii) Slope of the line parallel to y-axis = 0
y – y1 = m (x – x1)
and ⇒ 0 = x – x1
⇒ x + 5 = 0

#### Question 13. Equation Of A Line Concise Solutions

(i) Find the equation of the line passing through (5, -3) and parallel to x – 3y = 4.
(ii) Find the equation of the line parallel to the line 3x + 2y = 8 and passing through the point (0, 1). (2007)

(i) Writing the equation x – 3y = 4 in form of y = mx + c
-3y = -x + 4

⇒ 2y – 2 = -3x
⇒ 3x + 2y – 2 = 0
Which is the required equation.

#### Question 14.

Find the equation of the line passing through (-2, 1) and perpendicular to 4x + 5y = 6.

Writing the equation 4x + 5y = 6 in form of y = mx + c

#### Question 15.

Find the equation of the perpendicular bisector of the line segment obtained on joining the points (6, -3) and (0, 3).

#### Answer 15 Equation Of A Line Concise Solutions

The perpendicular of the line segment bisects it.
Co-ordinates of mid-point of the line segment which is obtained by joining the points (6, -3) and (0, 3)

Slope of the line perpendicular to it = 1 (Product of slopes = -1)
Equation of the perpendicular bisector is y – y1 = m (x – x1)
and y – 0 = 1 (x – 3)
y = x – 3

#### Question 16.

In the following diagram, write down:
(i) the co-ordinates of the points A, Band C.
(ii) the equation of the line through A and parallel to BC.

(i) From the figure, the see that co-ordinates of A are (2, 3), B are (-1, 2) and C are (3, 0)

#### Question 17.

B (-5, 6) and D (1, 4) are the vertices of rhombus ABCD. Find the equation of diagonal BD and of diagonal AC.

#### Question 18.

A = (7, -2) and C = (-1, – 6) are the vertices of a square ABCD. Find the equation of the diagonals AC and BD.

#### Question 19. Equation Of A Line Concise Solutions

A (1, -5), B (2, 2) and C (-2, 4) are the vertices of the ∆ABC, find the equation of:
(i) the median of the triangle through A.
(ii) the altitude of the triangle through B.
(iii) the line through C and parallel to AB.

(i) Let D be the mid-point of BC
co-ordinates mid-point of

#### Question 20.

(i) Write down the equation of the line AB, through (3, 2) and perpendicular to the line 2y = 3x + 5.
(ii) AB meets the x-axis at A and the y-axis at B. Write down the co-ordinates of A and B. Calculate the area of triangle OAB, where O is the origin. [1995]

(i) Write the equation 2y = 3x + 5 in the form of y = mx + c. We get:

y – 2 = $\frac { -2 }{ 3 }$ (x – 3)
⇒ 3y – 6 = -2x + 6
and ⇒ 2x + 3y = 6 + 6
⇒ 2x + 3y = 12 …… (i)

##### (ii)

AB meets the x-axis at A
ordinate (y) of A = 0 i.e. y = 0
Substituting, the value of y in (i)
2x + 3 x 0 = 12
⇒ 2x = 12
⇒ x = 6
Co-ordinates of A are (6, 0)
Again. AB meets y-axis at B
Abscissa of B = 0 i.e. x = 0
Substituting the value of x in (i)
2 x 0 + 3y = 12
⇒ y = 4
Co-ordinates of B are (0, 4)
Area of ∆OAB = $\frac { 1 }{ 2 }$ x Base x altitude
$\frac { 1 }{ 2 }$ x 4 x 6 = 12 square units

#### Question 21. Equation Of A Line Concise Solutions

The line 4x – 3y + 12 = 0 meets the x-axis at A. Write the co-ordinates of A.
Determine the equation of the line through A and perpendicular to 4x – 3y + 12 = 0.

The line 4x – 3y + 12 = 0 meets x-axis at A.
Ordinates of A = 0. i.e. y = 0
Substituting, the value of y in the equation
4x – 3 x 0 + 12 = 0
⇒ 4x + 12 = 0
and ⇒ 4x = -12
hence ⇒ x = -3
Co-ordinates of A are (-3, 0)
Writing the equation 4x – 3y + 12 = 0 in form of y = mx + c
⇒ -3y = -4x – 12

#### Question 22.

The point P is the foot of perpendicular from A (-5, 7) to the line is 2x – 3y + 18 = 0. Determine:
(i) the equation of the line AP.
(ii) the co-ordinates of P.

Write the equation in form of y = mx + c
2x – 3y + 18 = 0
⇒ -3y = -2x – 18
⇒ y = $\frac { 2 }{ 3 }$ x + 6 (Dividing by 3)
Slope of the line = $\frac { 2 }{ 3 }$
and slope of the line perpendicular to it = $\frac { -3 }{ 2 }$ (Product of slopes = -1)

##### (i)

Equation of line AP perpendicular to the given line and drawn through A (-5, 7)

y – y1 = m (x – x1)
⇒ y – 7 = $\frac { -3 }{ 2 }$ (x + 5)
⇒ 2y – 14 = -3x – 15
and ⇒ 3x + 2y – 14 + 15 = 0
hence ⇒ 3x + 2y + 1 = 0

##### (ii)

P is the point of intersection of these lines
we will solve their equations
2x – 3y + 18 = 0 ….(i)
3x + 2y + 1 = 0 ….(ii)
Multiplying (i) by 2 and (ii) by 3, we get
4x – 6y + 36 = 0
9x + 6y + 3 = 0
13x + 39 = 0
and ⇒ 13x = -39
⇒ x = -3
Now, substituting the value of x in (i)
2(-3) – 3y + 18 = 0
⇒ -6 – 3y + 18 = 0
and ⇒ -3y + 18 – 6 = 0
so ⇒ -3y + 12 = 0
hence⇒ -3y = -12
⇒ 3y = 12
⇒ y = 4
Co-ordinates of P are (-3, 4)

#### Question 23

The points A, B and C are (4, 0), (2, 2) and (0, 6) respectively. Find the equations of AB and BC. If AB cuts the y-axis at P and BC cuts the x-axis at Q, find the co-ordinates of P and Q.

AB meets y- axis at P
abscissa of P = 0 i. e. x = 0
Substituting the value of y in (i)
0 + y = 4
⇒ y = 4
Co-ordinates of P are (0, 4)
BC meets x-axis at Q
ordinate of Q = 0 i.e. y = 0
Substituting, the value of y in (ii),
2x + 0 = 6
⇒ 2x = 6
⇒ x = 3
Co-ordinates of Q are (3, 0)

#### Question 24

Match the equations A, B, C, and D with the lines L1, L2, L3 and L4, whose graphs are roughly drawn in the given diagram.
A = y = 2x;
B = y – 2x + 2 = 0;
C = 3x + 2y = 6;
D = y = 2 [1996]

A → L3,
B → L4,
C → L2,
D → L1

#### Question 25.

Find the value of ‘a’ for which the following points A (a, 3), B (2, 1) and C (5, a) are collinear. Hence find the equation of the line. (2014)

A (a, 3), B (2,1) and C (5, a) are collinear.
Slope of AB = Slope of BC

### EXERCISE – 14 (E) Solutions of Concise Selina Publications Maths Equation of a Straight Line

#### Question 1.

Point P divides the line segment joining the points A (8, 0) and B (16, -8) in the ratio, 3 : 5. Find its co-ordinates of point P. Also, find the equation of the line through P and parallel to 3x + 5y = 7

#### Question 2

The line segment joining the points A (3, -4) and B (-2, 1) is divided in the ratio 1 : 3 at point P in it Find the co-ordinates of P. Also, find the equation of the line through P and perpendicular to the line 5x – 3y = 4.

Point P, divides the line segment A (3, -4) and B(-2, 1) in the ratio of 1 : 3
Let co-ordinates of P be (x, y), then

#### Question 3

Question 3.

A line 5x + 3y + 15 = 0 meets y -axis at point P. Find the co-ordinates of point P. Find the equation of a line through P and perpendicular to x – 3y + 4 = 0.

P lies on y-axis and let the co-ordinates of P be (0, y)
P lies also on the line 5x + 3y + 15 = 0 it will satisfy it.
5 x 0 + 3y + 15 = 0
⇒ 3y = -15
⇒ y = -5
Co-ordinates of P are (0, -5)
Now, writing the line x – 3y + 4 = 0 is form of y = mx + c
-3y = -x – 4
and ⇒ 3y = x + 4
⇒ y = $\frac { 1 }{ 3 }$ x + $\frac { 4 }{ 3 }$

#### Question 4

Find the value of k for which the lines kx – 5y + 4 = 0 and 5x – 2y + 5 = 0 are perpendicular to each other. [2003]

Writing, the line kx – 5y + 4 = 0 in form of y = mx + c
⇒ -5y = -kx – 4
⇒ 5y = kx + 4

#### Question 5

A straight line passes through the points P (-1, 4) and Q (5, -2). It intersects the co-ordinate axes at points A and B. M is the mid point of the segment AB. Find:
(i) the equation of the line.
(ii) the co-ordinates of A and B.
(iii) the co-ordinates of M. (2003)

(ii) Let. co-ordinates of A be (x, 0) and of B be (0, y) which lie On the line.
Substituting, the co-ordinates in (i)
x + 0 = 3 ⇒x = 3
Co-ordinates of A are (3, 0)
Again 0 + y = 3 ⇒ y = 3
Co-ordinates of B are (0,3)
(iii) M is the mid-point of AB.
Co-ordinates of M wil be ($\frac { 3 }{ 2 }$ , $\frac { 3 }{ 2 }$)

#### Question 6

(1, 5) and (-3, -1) are the co-ordinates of vertices A and C respectively of rhombus ABCD. Find the equations of the diagonals AC and BD.

Co-ordinates of A and C of rhombus ABCD are (1, 5) and (-3, -1)

⇒ 3y – 6 = -2x – 2
and ⇒ 2x + 3y = 6 – 2
so ⇒ 2x + 3y = 4

#### Question 7

Show that A (3, 2), B (6, -2) and C (2, -5) can be the vertices of a square.
(i) Find the co-ordinates of its fourth vertex D, if ABCD is a square.
(ii) Without using the co-ordinates of vertex D, find the equation of side AD of the square and also the equation of diagonal BD.

Three vertices of a square ABCD are A (3, 2), B (6, -2) and C (2, -5)
Let, co-ordinates of fourth vertex D be (x, y)

#### Question 8

.A line through origin meets the line x = 3y + 2 at right angles at point X. Find the co-ordinates of X.

Slope of line x = 3y + 2 or 3y = x – 2 ….(i)

#### Question 9

A straight line passes through the point (3, 2) and the portion of this line intercepted between the positive axes, is bisected at this point. Find the equation of the line.

Let, the line intersects x-axis at A and y-axis at B.
Let, co-ordinates of A (x, o) and of (o, y)
But (3, 2) is the mid-point of AB.

#### Question 10

Find the equation of the line passing through the point of intersection of 7x + 6y = 71 and 5x – 8y = -23; and perpendicular to the line 4x -2y = 1.

7x + 6y = 71 ….(i)
5x – 8y = -23 ….(ii)
Multiply (i) by 4 and (ii) by 3,
28x + 24y = 284
15x – 24y = -69
On adding (i) and (ii), we get:
43x = 215
x = 5
Substituting, the value of x in (i)
7 x 5 + 6y = 71
35 + 6y = 71
⇒ 6y = 71 – 35 = 36
⇒ y = 6
Point of intersection of these lines is (5, 6)
Now slope of line 4x – 2y = 1
⇒ 4x – 1 = 2y
⇒ y = 2x – $\frac { 1 }{ 2 }$ is 2
Slope of line through the point of intersection and perpendicular to 4x – 2y = 1 is $\frac { -1 }{ 2 }$
Equation of the line y – y1 = m (x – x1)
⇒ y – 6 = $\frac { -1 }{ 2 }$ (x – 5)
and ⇒ 2y – 12 = -x + 5
hence ⇒ x + 2y = 5 + 12 = 17
⇒ x + 2y = 17

Question 11

Find the equation of the line which is perpendicular to the line $\frac { x }{ a }$ – $\frac { y }{ b }$ = 1 at the point where this line meets y-axis.

#### Question 12

O (0, 0), A (3, 5) and B (-5, -3) are the vertices of triangle OAB. Find:
(i) the equation of median of ∆OAB through vertex O.
(ii) the equation of altitude of ∆OAB through vertex B.

(i) Let, mid-point of AB be D.

#### Question 13

Determine whether the line through points (-2, 3) and (4, 1) is perpendicular to the line 3x = y + 1.
Does line 3x = y + 1 bisect the line segment joining the two given points ?

Slope of the line joining the points (-2, 3) and (4, 1)

Yes, these are perpendicular to each other
Let P be the mid-point of the line joining the points (-2, 3) and (4, 1)
Co-ordinates of P will be

This point (1, 2) satisfies the equaion 3x = y + 1 then, it will bisect the line joining the given point
now, substituting the value of x and y. in 3x = y + 1
⇒ 3 x 1 = 2 + 1
⇒ 3 = 3. which is true.
Yes, the line 3x = y + 1 is the bisector.

#### Question 14

Given a straight line x cos 30° + y sin 30° = 2. Determine the equation of the other line which is parallel to it and passes through (4, 3).

Equation of the given line is x cos 30° + y sin 30° = 2
y sin 36° = -x cos 30° + 2

#### Question 15

Find the value of k such that the line (k – 2) x + (k + 3) y – 5 = 0 is :
(i) perpendicular to the line 2x – y + 7 = 0
(ii) parallel to it.

Writing the given equation in the form of y = mx + c
(k – 2) x + (k + 3) y – 5 = 0 ….(i)
⇒ (k + 3) y = – (k – 2) x + 5

#### Question 16

The vertices of a triangle ABC are A (0, 5), B (-1, -2) and C (11, 7) write down the equation of BC. Find :
(i) the equation of line through A and perpendicular to BC.
(ii) the co-ordinates of the point P, where the perpendicular through A, as obtained in (i) meets BC.

Vertices of ∆ABC are A (0, 5),B (-1, -2) and C (11, 7)

(ii) Let, the line through A meets BC in P
P is point of intersection of these two lines.
3x – 4y = 5 ……… (i)
4x + 3y = 15 …….. (ii)
On solving (i), (ii) we get
x = 3, y = 1
Co-ordinates of Pare (3, 1)

#### Question 17

From the given figure, find :
(i) the co-ordinates of A, B and C.
(ii) the equation of the line through A and parallel to BC. (2005)

(i) From the figure, we see that co-ordinates of A are (2, 3), of B are (-1, 2) of C and (3, 0)
(ii) Slope of line BC is (m)

⇒ x + 2y – 6 – 2 = 0
and ⇒ x + 2y – 8 = 0
hence ⇒ x + 2y = 8

#### Question 18

P (3, 4), Q (7, -2) and R (-2, -1) are the vertices of triangle PQR. Write down the equation of the median of the triangle through R. (2004)

Let (x, y) be the co-ordinates of M, the mid-point of PQ.

⇒ x + 2y – 8 = 0
⇒ x + 2y = 8

#### Question 19

A (8, -6), B (-4, 2) and C (0, -10) are the vertjces of a triangle ABC. If P is the mid-point of AR and Q is the mid-point of AC, use co ordinate geometry to show that PQ is parallel to BC. Give a special name to quadrilateral PBCQ.

In ∆ABC, co-ordinates of A, B and C are (8, -6), (-4, 2) and C (0, -10) respectively.
P and Q are the mid-points of AB and AC respectively
Co-ordinates of P will be

Slopes of PQ and BC are same.
These are parallel to each other.

#### Question 20

.A line AB meets the x-axis at point A and y-axis at point B. The point P(-4, -2) divides the line segment AB internally such that AP : PB = 1 : 2. Find :
(i) the co-ordinates of A and B.
(ii) equation of line through P, and perpendicular to AB.

Line AB intersects x-axis at A and y-axis at B.
(i) Let co-ordinates of A be (x, 0) and of B be (0, y)
Point P (-4, -2) intersects AB in the ratio 1 : 2

#### Question 21.

A line intersects x-axis at point (-2, 0) and cuts off an intercept of 3 units from die positive side of y-axis. Find the equation of the line. (1992)

Let line intersects x-axis at P (-2, 0) and cuts off an intercept of 3 units at Q.

#### Question 22.

Find the equation of a line passing through the point (2, 3) and having the x-intercept of 4 units. (2002)

x-intercept = 4
Co-ordinates of that point = (4, 0)
The co-ordinates of the given point (2, 3)

#### Question 23.

The given figure (not drawn to scale) shows two straight lines AB and CD. If’ equation of the line AB is : y = x + 1 and equation of line CD is : y = √3 x – 1. Write down the inclination of lines AB and CD; also, find the angle 6 between AB and CD. (1989)

Equation of line AB is y = x + 1
and equation of line CD is y = √3 x – 1
Slope of AB = 1
tanθ = 1
⇒ θ = 45°
Inclination angles of AB = 45°
Slope of CD = tanθ = √3 = tan 60°
⇒ θ = 60°
Inclination angle of CD = 60°
In ΔPQR,
Ext. ∠RQX = ∠RPQ + ∠PRQ (Exterior angles is equal to sum of its interior opposite angles)
⇒ 60° = 45° + θ
and ⇒ θ = 60° – 45° = 15°
hence ⇒ θ = 15°

#### Question 24.

Write down the equation of the line whose gradient is $\frac { 3 }{ 2 }$ and which passes through P, where P divides the line segment joining A (-2, 6) and B (3, -4) in the ratio 2 : 3. (1996, 2001)

P divides the line segment AB in which A (-2, 6) and B (3, -4) in the ratio 2 : 3
Co-ordinates of P will be

#### Question 25.

The ordinate of a point lying on the line joining the points (6, 4) and (7, -5) is -23. Find the co-ordinates of that point.

Let points are A (6, 4) and B (7, -5)

#### Question 26.

Points A and B have coordinates (7, -3) and (1, 9) respectively. Find
(i) the slope of AB.
(ii) the equation of the perpendicular bisector of the line segment AB.
(iii) the value of ‘p’ of (-2, p) lies on it.

Coordinates of A are (7, -3), of B = ( 1, 9)

#### Question 27.

A and B are two points on the x-axis and y-axis respectively. P (2, -3) is the mid point of AB. Find the
(i) Coordinates of A and B.
(ii) Slope of line AB.
(iii) equation of line AB.

As P (2, -3) is mid-point of AB.
Let coordinates of B be (0, y) and coordinates of A be (x, 0)

#### Question 28.

The equation of a line is 3x + 4y – 7 = 0. Find:
(i) the slope of the line.
(ii) the equation of a line perpendicular to the given line and passing through the intersection of the lines x – y + 2 = 0 and 3x + y – 10 = 0.

Given line 3x + 4y -7 = 0

⇒ 3 (y – 4) = 4 (x – 2)
and ⇒ 3y – 12 = 4x – 8
so ⇒ 4x – 3y – 8 + 12 = 0
hence ⇒ 4x – 3y + 4 = 0

#### Question 29.

ABCD is a parallelogram where A (x, y), B (5, 8), C (4, 7) and D (2, -4). Find :
(i) co-ordinates of A
(ii) equation of diagonal BD.

(i) In || gm ABCD, A (x, y), B (5, 8), C (4, 7) and D (2, -4)
The diagonals of ||gm bisect each other
O is said point of AC and BD
Now if O is mid point of BD then its co-ordinates will be

⇒ y + 4 = 4x – 8
and ⇒ 4x – y -8 – 4 = 0
hence ⇒ 4x – y – 12 = 0 or 4x – y = 12

#### Question 30.

Given equation of line L1 is y = 4.
(i) Write the slope of line L2 if L2 is the bisector of angle O.
(ii) Write the co-ordinates of point P.
(iii) Find the equation of L2.

(i) Equation of line L1 is y = 4
L2 is the bisector of ∠O

#### Question 31.

Find:

(i) equation of AB
(ii) equation of CD

Co-ordinates of A and B are (-5, 4) and (3, 3) respectively.

#### Question 32.

Find the equation of the line that has x-intercept = -3 and is perpendicular to 3x + 5y = 1

x-intercept of the line = -3
and is perpendicular to the line
3x + 5y = 1
5y = 1 – 3x

#### Question 33.

A straight line passes through the points P (-1, 4) and Q (5, -2). It intersects x-axis at point A and y-axis at point B. M is the mid-point of the line segment AB. Find :
(i) the equation of the line.
(ii) the co-ordinates of points A and B.
(iii) the co-ordinates of point M.

A line passing through the two points P (-1, 4) and Q (5, -2) intersects x-axis at point A and y- axis at point B.
M is mid-point of AB

(i) Equation of line AB will be
y – y1 = m (x – x1)
⇒ y – 4 = -1 (x + 1)
and ⇒ y – 4 = – x – 1
therefore⇒ y + x = -1 + 4
hence ⇒ x + y = 3
(ii) The line intersect x-axis at A and OA = 3 units
Co-ordinates of A are (3, 0) and the line intersects y-axis at B and OB = 3 units
Co-ordinates of B are (0, 3)
(iii) M is mid-point of AB
Co-ordinates of M are ($\frac { 3 }{ 2 }$ , $\frac { 3 }{ 2 }$)

#### Question 34.

In the given figure, line AB meets y-axis at point A. Line through C (2, 10) and D intersects line AB at right angle at point P. Find:

(i) equation of line AB.
(ii) equation of line CD.
(iii) co-ordinates of points E and D.

In the given figure, AB meets y-axis at point A.
Line through C (2, 10) and D intersects line AB at P at right angle.

Equation of CD
y – 10 = 3 (x – 2)
⇒ y – 10 = 3x – 6
and ⇒ 3x – y + 10 – 6 = 0
so ⇒ 3x – y + 4 = 0
(iii) Co-ordinates of D which is on x-axis
3x – y + 4 = 0
3x – 0 + 4 = 0
⇒ 3x + 4 = 0
and ⇒ 3x = -4
hence ⇒ x = $\frac { -4 }{ 3 }$
Co-ordinates of D are ($\frac { -4 }{ 3 }$ , 0)
E is also on x-axis
x + 3y = 18
Substituting, y = 0, then
x + 0= 18
⇒ x = 18
Co-ordinates of E are (18, 6)

#### Question 35.

A line through point P (4, 3) meets x-axis at point A and the y-axis at point B. If BP is double of PA, find the equation of AB.

A line through P (4, 3) meets x-axis at A and the y-axis at B. If BP is double of PA.
Draw BC || x-axis
and PC || y-axis
∠P = ∠P (common)
∠D = ∠C (each 90°)
PB = 2PA ⇒ PA = $\frac { 1 }{ 2 }$ PB

2y – 6 = 3x – 12
⇒ 3x – 2y – 12 + 6 = 0
and ⇒ 3x – 2y – 6 = 0
hence ⇒ 3x – 2y = 6

#### Question 36.

Find the equation of line through the intersection of lines 2x – y = 1 and 3x + 2y = -9 and making an angle of 30° with positive direction of x-axis.

Equation of given two intersecting lines are 2x – y = 1 and 3x + 2y = -9 Which make an angle of 30°

#### Question 37.

Find the equation of the line through the points A (-1, 3) and B (0, 2). Hence, show that the points A, B and C (1, 1) are collinear.

The given points are A (-1, 3) and B (0, 2) and co-ordinates of a point C are (1, 1)
Now slope of the line joining A and B

Equation of the line y – y1 = m(x – x1)
⇒ y – 2 = -1 (x – 0)
and ⇒ y – 2 = -x
hence ⇒ x + y – 2 = 0
Point C (1, 1) will be on AB if it satisfy
1 + 1 – 2 = 0
⇒ 0 = 0
Point C lies on AB
Hence points A, C and B are collinear.

#### Question 38.

Three vertices of a parallelogram ABCD taken in order are A (3, 6), B (5, 10) and C (3, 2), find:
(i) the co-ordinates of the fourth vertex D.
(ii) length of diagonal BD.
(iii) equation of side AB of the parallelogram ABCD. (2015)

Three vertices of a ||gm ABCD taken an order are A (3, 6), B (5, 10) and C (3, 2)
Join diagonals AC and BD which bisect each other at O.
O is mid-point of AC as well as of BD
Now co-ordinates of O will be

#### Question 39.

In the figure, given, ABC is a triangle and BC is parallel to the y-axis. AB and AC intersect the y-axis at P and Q respectively.

(i) Write the co-ordinates of A.
(ii) Find the length of AB and AC.
(iii) Find the ratio in which Q divides AC.
(iv) Find the equation of the line AC. (2015)

In the given figure,
ABC is a triangle and BC || y-axis
AB and AC intersect the y-axis at P and Q respectively.
(i) Co-ordinates of A are (4,0).
(ii) Length of AB

#### Question 40.

The slope of a line joining P (6, k) and Q (1 – 3k, 3) is $\frac { 1 }{ 2 }$. Find :
(i) k
(ii) mid-point of PQ, using the value of ‘A’ found in (i). (2016)

(i) Slope of the line joining P(6, k) and Q (1 –

#### Question 41.

A line AB meets X-axis at A and Y-axis at B. P (4, -1) divides AB in the ratio 1 : 2.
(i) Find the coordinates of A and B.
(ii) Find the equation of the line through P and perpendicular to AB.

(i) Since, A lies on the x-axis,
let the coordinates of A be (x, 0).
Since B lies on the y-axis,
let the coordinates of B be (0, y).
Let m = 1 and n = 2.
Using section formula,

Thanks